Evolution Strategies — (μ,λ)-ES and (μ+λ)-ES

Evolution Strategies (ES) are among the oldest population-based optimizers, developed by Ingo Rechenberg and colleagues in 1960s Germany; the (μ+λ) variant came from Schwefel (and J. Reichenbacher) in the 1970s. They are the classical ancestors of cma-es. Source: andrey-dik‘s MQL5 implementation, with recombination letting offspring “inherit the characteristics of all parents.”

How it works — comma vs. plus

μ parents generate λ offspring via recombination + (self-adaptive) mutation; the two variants differ in who survives:

• (μ,λ)-ES (“comma”) — select the best μ from the λ offspring only; parents always die. New children “compete with their parents for a limited number of places, which can lead to premature convergence to a local optimum.”
• (μ+λ)-ES (“plus”) — select the best μ from parents ∪ offspring combined; good parents persist, giving “a broader exploration of the solution space.” A textbook exploration-vs-exploitation knob: elitist retention (plus) vs. forced turnover (comma).

Benchmark (context-relative)

Reported on the mature %-of-MAX rating (Hilly/Forest/Megacity) — the same scale as the founding four, so directly comparable:

(μ+λ)-ES achieved “phenomenal results” — “ahead of its nearest competitor SDSm by almost 10%”, and above all four founding algorithms (best of those: BSA at 55.10%). It led the suite from its 2026-06 ingest until 2026-07-25, when DOAm (73.63%) edged past it by 1.45 points. A 1970s elitist strategy holding the top of the modern field for that long is a sharp no-free-lunch-theorem twist: the “plus” tweak (keep parents) outperforms decades of newer metaphors, and the thing that finally beat it was a three-line correction to an existing metaphor algorithm rather than a new mechanism. The comma variant lands mid-pack, showing the survival rule alone swings rank by ~20 points.

Related

metaheuristic-optimization · exploration-vs-exploitation · population-optimization-benchmark · no-free-lunch-theorem · andrey-dik · cma-es · backtracking-search-algorithm